// Numbas version: finer_feedback_settings {"name": "Ex 5 Determinant and Inverse of 2x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b12": {"description": "", "name": "b12", "definition": "random(-5..5)", "group": "Matrix B", "templateType": "anything"}, "c21": {"description": "", "name": "c21", "definition": "random(2..5)", "group": "Matrix C", "templateType": "anything"}, "b22": {"description": "", "name": "b22", "definition": "random(-9..9 except [0,b21*b12/b11])", "group": "Matrix B", "templateType": "anything"}, "b11": {"description": "", "name": "b11", "definition": "random(1..9 except a11)\n//if(a11=tr2,tr2+1,tr2)", "group": "Matrix B", "templateType": "anything"}, "c": {"description": "", "name": "c", "definition": "matrix([\n [c11,c12],\n [c21,c22]\n])", "group": "Matrix C", "templateType": "anything"}, "c22": {"description": "", "name": "c22", "definition": "random(1..9 except c21*c12/c11)", "group": "Matrix C", "templateType": "anything"}, "b21": {"description": "", "name": "b21", "definition": "random(-6..6 except 0)", "group": "Matrix B", "templateType": "anything"}, "a21": {"description": "", "name": "a21", "definition": "random(-6..6 except 0) ", "group": "Matrix A", "templateType": "anything"}, "b": {"description": "", "name": "b", "definition": "matrix([\n [b11,b12],\n [b21,b22]\n])", "group": "Matrix B", "templateType": "anything"}, "a11": {"description": "", "name": "a11", "definition": "random(-9..9 except 0)", "group": "Matrix A", "templateType": "anything"}, "a12": {"description": "", "name": "a12", "definition": "random(-5..5)", "group": "Matrix A", "templateType": "anything"}, "a": {"description": "", "name": "a", "definition": "matrix([\n [a11,a12],\n [a21,a22]\n])", "group": "Matrix A", "templateType": "anything"}, "c12": {"description": "", "name": "c12", "definition": "a12+b12", "group": "Matrix C", "templateType": "anything"}, "a22": {"description": "", "name": "a22", "definition": "random(1..9 except a21*a12/a11)", "group": "Matrix A", "templateType": "anything"}, "c11": {"description": "", "name": "c11", "definition": "random(1,2,4)", "group": "Matrix C", "templateType": "anything"}}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {"inverse": {"language": "jme", "parameters": [["m", "matrix"]], "definition": "matrix([\n [m[1][1],-m[0][1]],\n [-m[1][0],m[0][0]]\n])/det(m)", "type": "matrix"}}, "metadata": {"description": "

Find the determinant and inverse of three $2 \\times 2$ invertible matrices.

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Determinant of a $2 \\times 2$ matrix

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The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is given by

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\\[ \\det\\left(\\mathrm{M}\\right) = ad-bc \\]

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If we have two $n \\times n$ matrices $M$ and $N$, then

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\\[ \\det\\left(\\mathrm{MN}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right) \\]

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And it follows that if we have a third matrix $P$,

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\\[ \\det\\left(\\mathrm{MNP}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right)\\det\\left(\\mathrm{P}\\right) \\]

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a)

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Thus for our example we have:

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\\begin{align}
\\det\\left(\\mathrm{A}\\right) &= \\simplify[]{{a11}*{a22}-{a12}*{a21} = {det(a)}} \\\\
\\det\\left(\\mathrm{B}\\right) &= \\simplify[]{{b11}*{b22}-{b12}*{b21} = {det(b)}} \\\\
\\det\\left(\\mathrm{C}\\right) &= \\simplify[]{{c11}*{c22}-{c12}*{c21} = {det(c)}}
\\end{align}

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\\begin{align}
\\det\\left( \\mathrm{ABC} \\right) &= \\det(\\mathrm{A}) \\det(\\mathrm{B}) \\det(\\mathrm{C}) \\\\
&= \\simplify[]{{det(a)}*{det(b)}*{det(c)}} \\\\
&= \\var{det(a*b*c)}
\\end{align}

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Inverse of a $2 \\times 2$ matrix

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Suppose $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is a $2 \\times 2$ matrix and $\\det\\left(\\mathrm{M}\\right) = \\Delta \\neq 0$.

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Then $\\mathrm{M}$ is invertible and

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\\[ \\mathrm{M}^{-1} = \\frac{1}{\\Delta} \\begin{pmatrix} d & -b\\\\ -c& a \\end{pmatrix}=\\begin{pmatrix} \\frac{d}{\\Delta} & -\\frac{b}{\\Delta}\\\\ -\\frac{c}{\\Delta}& \\frac{a}{\\Delta} \\end{pmatrix}\\]

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Applying this to these examples we obtain:

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b)

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\\[ \\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}} \\]

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c)

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\\[ \\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}} \\]

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d)

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\\[ \\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}} \\]

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Do the following matrix problems.

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Let

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\\[\\mathrm{A} = \\var{a},\\;\\; \\mathrm{B} = \\var{b},\\;\\; \\mathrm{C} = \\var{c}\\]

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Calculate the determinants of these matrices:

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$\\det\\left(\\mathrm{A}\\right) = $ [[0]]

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$\\det\\left(\\mathrm{B}\\right) = $ [[1]]

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$\\det\\left(\\mathrm{C}\\right) = $ [[2]]

\n

$\\det\\left(\\mathrm{ABC}\\right) = $ [[3]]

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Find the inverses of the matrices given above. Input all matrix entries as fractions or integers and not as decimals.

\n

$\\mathrm{A}^{-1} = $ [[0]]

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$\\mathrm{B}^{-1} = $ [[0]]

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$\\mathrm{C}^{-1} = $ [[0]]

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